## Dept. of Mathematics and Computer Science (Berlin)

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**reviewer:** Miloslav Znojil
**reviewernum:** 9689

**revieweremail:** znojil@ujf.cas.cz

**zblno:** DE015197664

**author:** Bojanczyk, A. W.; Lutoborski, A.:

**shorttitle:** Procrustes problem for orthogonal Stiefel matrices

**source:** SIAM J. Sci. Comput. 21, No. 4, 1291 - 1304 (2000)

**rpclass:** 49M20

**rsclass:** 65K10; 51F99

**keywords:** minimizing distance between a point and an ellipsoid, Stiefel manifolds, projections, relaxation methods, Procrustes problem

**revtext:**
Two m-dimensional vectors B and B' define an ellipse E(t) = B cos
t + B' sin t. Its distance from a point A is given by the vector
A-E(t') of minimal length. The Procrustes problem is to find the
``angle" t'. One can generalize of course, taking p < m different
vectors B and replacing the two-dimensional vector (cos t, sin t)
of unit length by k < p columns of a p x p orthogonal matrix (the
latter k-plets form a set OSt(p,k) of the so called orthogonal
Stiefel matrices).
The paper proposes a class of relaxation methods for generating
sequences of approximations to the k(p-(k+1)/2)-parameteric
Stiefel-matrix minimizer. A numerical illustration and geometric
interpretation of these methods is offered.

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